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Russell Jay Hendel
Linear equalities in Fibonacci numbers,
Fibonacci Quart. 44 (2006), no. 3, 235–248.
Abstract
We study the equation Fb = Fx(1) + F−x(3) + F−x(4) + · · · + F−x(m)
with m ≥ 3, 0 < x(1) < b, and 0 < x(3) < x(4) < · · · < x(m). This
equation naturally arises in the generalization of several problems that
have appeared in The Fibonacci Quarterly in the problem sections.
This equation also has intrinsic interest in its own right. The main
theorem - the Accident theorem–states, that under very mild conditions, solutions to this equation cannot happen by accident; that is,
there are no singular solutions, but rather every solution belongs to a
parametrizable class of solutions. Furthermore if m ≥ 4, then b must
be even and there are exactly 9 parametrizable solutions. This is the
first major theorem in the literature on identities in Fibonacci numbers with an arbitrary number of summands whose subscripts have
mixed signs. There are only 2 hypothesii of the accident theorem: We
require that for all i, x(i) > 2 and that no proper subset of summands on the right hand side of the equation has a sum of zero. While
the proof of the accident theorem requires many sub-theorems and
lemmas, the basic proof method is to exploit Fibonacci Telescoping
lemmas. An example of Fibonacci Telescoping is illustrated by the following identity which leads to one of the 9 parametrizable solutions for
m ≥ 4 : Fb = Fb−o−3 + F−(b−o−2) + F−(b−o) + · · · + F−(b−1) with b even
and o an arbitrary positive odd integer.
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